To prepare us for the amazing proof that 2+2 makes 5, let's sample some "Classic Jokes about Mathematicians" first...

...Engineer: "How can you POSSIBLY visualize something that occurs in 9-dimensional space?" Mathematician: "Easy, first visualize it in N-dimensional space, then let N go to 9"

...once they locked a mathematician in a cell with plenty of canned food and water but no can opener. Eventually they found his dessicated corpse, which was propped calmly against a wall, among piles of unopened cans and this was inscribed on the floor in blood: Theorem: If I can't open these cans, I'll die. Proof: assume the opposite...

...Theorem: All horses have an infinite number of legs. Proof: Everyone would agree that all horses have an even number of legs. It is also well-known that horses have forelegs in front and two legs in back. 4 + 2 = 6 legs, which is certainly an odd number of legs for a horse to have! Now the only number that is both even and odd is infinity; therefore all horses have an infinite number of legs.

...The biologist : "Look! There's a herd of zebras! And there, in the middle : A white zebra! It's fantastic ! There are white zebra's ! We'll be famous !" The statistician : "It's not significant. We only know there's one white zebra." The mathematician : "Actually, we only know there exists a zebra, which is white on one side." The computer scientist : "Oh, no! An Exception Occured..!"

...The USDA once wanted to make cows produce milk faster, to improve the dairy industry. The foremost mathematician of his time offered to help them with the problem. Upon hearing the problem, he told the delegation that they could come back in the morning and he would have solved the problem. In the morning he handed them a piece of paper with the computations for the new, 300% improved milk cow. The plans began: "A Proof of the Attainability of Increased Milk Output from Bovines: Consider a spherical cow......"

...A mathematician is a person who says that, when 3 people are supposed to be in a room but 5 came out, 2 more have to go in so the room gets empty...

...There was the following scrawled on a math office blackboard in college: 1 + 1 = 3, for large values of 1

"First and above all he was a logician. At least thirty-five years of the half-century or so of his existence had been devoted exclusively to proving that two and two always equal four, except in unusual cases, where they equal three or five, as the case may be."-- Jacques Futrelle, "The Problem of Cell 13"

Most mathematicians are familiar with -- or have at least seen references in the literature to -- the equation 2 + 2 = 4. However, the less well known equation 2 + 2 = 5 also has a rich, complex history behind it. Like any other complex quantitiy, this history has a real part and an imaginary part; we shall deal exclusively with the latter here.

Many cultures, in their early mathematical development, discovered the equation 2 + 2 = 5. For example, consider the Bolb tribe, descended from the Incas of South America. The Bolbs counted by tying knots in ropes. They quickly realized that when a 2-knot rope is put together with another 2-knot rope, a 5-knot rope results.

Recent findings indicate that the Pythagorean Brotherhood discovered a proof that 2 + 2 = 5, but the proof never got written up. Contrary to what one might expect, the proof's nonappearance was not caused by a cover-up such as the Pythagoreans attempted with the irrationality of the square root of two. Rather, they simply could not pay for the necessary scribe service. They had lost their grant money due to the protests of an oxen-rights activist who objected to the Brotherhood's method of celebrating the discovery of theorems. Thus it was that only the equation 2 + 2 = 4 was used in Euclid's "Elements," and nothing more was heard of 2 + 2 = 5 for several centuries.

Around A.D. 1200 Leonardo of Pisa (Fibonacci) discovered that a few weeks after putting 2 male rabbits plus 2 female rabbits in the same cage, he ended up with considerably more than 4 rabbits. Fearing that too strong a challenge to the value 4 given in Euclid would meet with opposition, Leonardo conservatively stated, "2 + 2 is more like 5 than 4." Even this cautious rendition of his data was roundly condemned and earned Leonardo the nickname "Blockhead." By the way, his practice of underestimating the number of rabbits persisted; his celebrated model of rabbit populations had each birth consisting of only two babies, a gross underestimate if ever there was one.

Some 400 years later, the thread was picked up once more, this time by the French mathematicians. Descartes announced, "I think 2 + 2 = 5; therefore it does." However, others objected that his argument was somewhat less than totally rigorous. Apparently, Fermat had a more rigorous proof which was to appear as part of a book, but it and other material were cut by the editor so that the book could be printed with wider margins.

Between the fact that no definitive proof of 2 + 2 = 5 was available and the excitement of the development of calculus, by 1700 mathematicians had again lost interest in the equation. In fact, the only known 18th-century reference to 2 + 2 = 5 is due to the philosopher Bishop Berkeley who, upon discovering it in an old manuscript, wryly commented, "Well, now I know where all the departed quantities went to -- the right-hand side of this equation." That witticism so impressed California intellectuals that they named a university town after him.

But in the early to middle 1800's, 2 + 2 began to take on great significance. Riemann developed an arithmetic in which 2 + 2 = 5, paralleling the Euclidean 2 + 2 = 4 arithmetic. Moreover, during this period Gauss produced an arithmetic in which 2 + 2 = 3. Naturally, there ensued decades of great confusion as to the actual value of 2 + 2. Because of changing opinions on this topic, Kempe's proof in 1880 of the 4-color theorem was deemed 11 years later to yield, instead, the 5-color theorem. Dedekind entered the debate with an article entitled "Was ist und was soll 2 + 2?"

Frege thought he had settled the question while preparing a condensed version of his "Begriffsschrift." This condensation, entitled "Die Kleine Begriffsschrift (The Short Schrift)," contained what he considered to be a definitive proof of 2 + 2 = 5. But then Frege received a letter from Bertrand Russell, reminding him that in "Grundbeefen der Mathematik" Frege had proved that 2 + 2 = 4. This contradiction so discouraged Frege that he abandoned mathematics altogether and went into university administration.

Faced with this profound and bewildering foundational question of the value of 2 + 2, mathematicians followed the reasonable course of action: they just ignored the whole thing. And so everyone reverted to 2 + 2 = 4 with nothing being done with its rival equation during the 20th century. There had been rumors that Bourbaki was planning to devote a volume to 2 + 2 = 5 (the first forty pages taken up by the symbolic expression for the number five), but those rumor remained unconfirmed. Recently, though, there have been reported computer-assisted proofs that 2 + 2 = 5, typically involving computers belonging to utility companies. Perhaps the 21st century will see yet another revival of this historic equation.

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5 Comments:

And if you ever stumble across an infinite legged horse, a spherical cow, a cat with nine tails, and a white zebra, you can be pretty sure you have entered the realm of the mathemagician from "The Phantom Tollbooth".

"...The USDA once wanted to make cows produce milk faster, to improve the dairy industry. The foremost mathematician of his time offered to help them with the problem. Upon hearing the problem, he told the delegation that they could come back in the morning and he would have solved the problem. In the morning he handed them a piece of paper with the computations for the new, 300% improved milk cow.

The plans began:A Proof of the Attainability of Increased Milk Output from Bovines: Consider a spherical cow....."

In similar vein to "2+2=5", here's something I wrote a while back that you might like. It outlines the problems inherent in the number 2, in light of Grabel's Law:"2 is not equal to 3 -- even for very large values of 2."